Normal Modal Logics

Introduction

I compared the <> operator to the $ operator and the [] operator to ". 

The similarity continues into the tree rules that apply to wffs headed by each operator. 

We saw that <>S5 introduced a world index that hadn't been used above in that branch, just like EI introducing a constant. And []S5 could introduce any world index at all just as UI could for constants.

We call <>S5 a world-generator rule and []S5 a world-filler. []S5 allows us to say that certain formulas are true in a particular world.

With the rules that were presented previously for a model modal logic, there was no distinction amongst the worlds introduced, and so we got the particular logic S5 (hence the labels for the rules.) But that's not always the case. We can get different logics by introducing relationships amongst the worlds. Kripke had a relationship of accessibility in his system - so that some worlds were accessible from other worlds and others weren't and this made a difference to the semantic values of modal statements with respect to those worlds. We'll introduce something of the same sort here.

Accessibility

If we have <>p in world n, then an application of <>S5 will give us p in world k, say.

In this case n has generated k.

You can see this in the tree to prove the contingency of ((<>p & <>q) É <>(p & q))

We could restrict the filler rule so that []p in n yields p in k only when n generates k. 

This generation is also called access. n generates k is the same as saying n has access to k.

Note that if n generates k and n generates l, we know nothing about the accessibility of k from l or v.v. 

We'll need to track the accessibility relations as they are created in our trees. So introduce these rules:

<>R                 <>X                w/v      

                            ...                              

                          wAv

                            X                  v

                         where v is new to this path

[]R                     []X                w\v      

                          wAv

                           ...                              

                            X                  v

Note:

1.  []R doesn't allow []X to yield X on any world, only on the worlds marked as accessible to the world in which []X holds.

2  worlds don't necessarily have access to themselves. S5 is special in this respect, and the rule []T is also a way of getting the result of self access.

Try                  []p É [][] p with just the rules of standard propositional logic, MN, []R, and <>R.

                         We find that it doesn't close, because n doesn't have access to the world in which ~P holds

We call the system with just those rules K

Try                  p É []<>p with just the rules of K.

                         We find that it doesn't close either

K to S5

K and S5 are normal modal logics. 

S5 is clearly a modification of K which can also be modified to yield other normal logics called T, S4, and Br.

There are two ways to make these modifications, either by adding new world-filler rules (Hintikka Strategy) or by declaring different properties for the accessibility relation (Orthodox Strategy - see Kripke.)

Hintikka Strategy                                                                                            

We need to be able to talk about the various combinations of tree rules. Here are some abbreviations:

PTr = tree rules for standard propositional logic

MN = Modal negation rules (see above)

KTr = PTr + MN + {<>R, []R}

S5Tr = PTr + MN + {<>S5, []S5, []T}

K to T                                                   

The system T is constructed by adding []T to the rules for K. Thus 

TTr = KTr + []T

As we modify the semantic rules for the modal logics we change the formulas that are going to be tautological in the system. There are certain formulas that are important in this respect.

Try                  []p É p in K.

                         We find that it closes in T but not in K.

T to S4                                                 

Add the following rule to T

[][]R                   []X                w\v      

                           wAv

                           ...                              

                            []X                v

S4Tr = TTr + [][]R

Try                  []p É [][]p in T.

                         We find that it closes in S4 but not in T.

S4 to S5                                               

Add the following rule to T

[][]SymR           []X                v      

                           wAv

                           ...                              

                            []X                w

S5Tr = S4Tr + [][]SymR

Try                  <>p É []<>p

                         We find that it closes in S5 but not in S4.

Orthodox Strategy                                                                                           

To distinguish the orthodox rule sets from the Hintikka, we'll prefix them with O.  

K to T                                                   

The system T was constructed above by adding []T to the rules for K. We get the equivalent effect by making every world accessible to itself. This is achieved by making the accessibility relation reflexive. A rule to specify this is

Refl                     ...                              

                           wAw

                           for any w on this branch

The system T is alternatively constructed by adding Refl to the rules for K. Thus 

OTTr = KTr + Refl

OTTr = PTr + MN + {<>R, []R, Refl}

Try again       []p É p

                         We find that it closes in T but not in K.

T to S4                                                 

Add the property of transitivity to the accessibility relation in T

Trans                wAv

                           vAu

                           ...                              

                           wAu

OS4Tr = OTTr + Trans

Try                  []p É [][]p

                         We find that it closes in S4 but not in T.

T to Br                                                  

Add the property of Symmetry to the accessibility relation in T

Sym                   wAv

                           ...                              

                           vAw

OBrTr = OTTr + Sym

Try                  p É []<>p

                         We find that it closes in Br but not in T.

S4 or Br to S5                                    

Give the accessibility relation all the named properties (and make it an equivalence relation)

OS5Tr = PTr + MN + {<>R, []R, Refl, Trans, Sym}

Try                  <>p É []<>p in S4.

                         We find that it closes in S5 but not in S4 or Br.

The Normal logics are related thus:

                                  Br   Þ   S5 

                                   Ý            Ý

                     K    Þ   T    Þ   S4 

Counter-examples

We can use much the same system of discovering counterexamples as we did previously, using the truth tree that we create to demonstrate failure of validity or tautology in some modal system. Or we can do something like a MAV exercise. We'll look at both methods, but in either case we'll need to make some modifications to the techniques to take account of the modifications made to the world-filling tree rules or to the accessibility relation. 

In fact we'll see that the most straightforward way to make the proper modification is to refer directly to the accessibility relationships.

Try                  []p É p in K as before

We get the tree

                        ~([]p É p)  n

                                 []p      n

                                 ~p       n

which is as far as we can go. The tree doesn't close so all the wffs in the open branch can be true together. 

So []p is true in n and yet p is false in n, and n is the only world that is supposed. 

The point to note is that there are no accessible worlds from n according to this tree.

Given that []p in n is true iff for every world k, if k is accessible from n then p is true in k, it follows (as there are no k accesssible from n,) that the condition holds and []p is true in n.

We write this counter-example as 

 

	n
p	0
[]p	1

Check that []p É p is false in n. It is.

Try                  (<>p & <>q) É <>(p & q) in K as before.

From the tree that we get we can derive values for p and q in worlds n, k, l.

Note that here we also have accessibility relations generated. 

We note all these things in a table:

 

	nAk, nAl
	n	k	l
p	0	1	0
q	0	0	1
p&q	0	0	0
<>p	1	0	0
<>q	1	0	0
<>(p&q)	0	0	0

Then calculate the value for the formula in this example.

Try                   <>p É []<>p in T.

The tree we got gives us 

 

	nAk, nAl, nAn, kAk, lAl
	n	k	l
p		0	1
<>p	1	0
[]<>p	0	0

Note the accessibility info in the top row and the use of the reflexive property of T's accessibility relation 

Now we'll try something more like MAV.

Try                   []<>p É <>[]p in T

The tree we get, which doesn't close in T, generates 7 worlds. But it's easier to try to generate a counterexample by assigning values. Starting with the simplest possibility, we find that a singleton universe will not work, so we try a universe with just 2 worlds in it.

Assume nAk and note that Refl(A) because this is T after all.

Assume the wff is false. Then:

 

	nAk nAn kAk
	n	k
p
<>p
[]<>p	1
[]p
<>[]p	0

We have nAn and nAk so for []<>p to be true in n we need <>p true in n and k.

Similarly, for <>[]p to be false in n we need []p is false in n and k. So:

 

	nAk nAn kAk
	n	k
p
<>p	1	1
[]<>p	1
[]p	0	0
<>[]p	0

For []p false in k, and kAk, we need p false in k. Thus:

 

	nAk nAn kAk
	n	k
p		0
<>p	1	1
[]<>p	1
[]p	0	0
<>[]p	0

But now note that this means that <>p in k must be false because we only have kAk, and not kAn.

Now here's the tricky part, we can introduce kAn, and make A symmetric. That would allow us to keep <>p as true if we also set p as true in n. So:

 

	nAk nAn kAk, kAn
	n	k
p	1	0
<>p	1	1
[]<>p	1
[]p	0	0
<>[]p	0

And this is a good counterexample in a T system (which also happens to be a Br system, but that's not a problem.)

Non-Reflexive Normal Modal Logics

The Normal logics are related thus:

                                            Br         Þ         S5

                                                    (+Trans)

                                             Ý                         Ý

                                        (+Sym)               (+Sym)

                     K        Þ         T          Þ          S4

                            (+Refl)            (+Trans)

But suppose that we didn't add Refl to K and we skipped T altogether. This is what we'd get:

                                            KB         Þ         K5

                                                    (+Trans)

                                             Ý                         Ý

                                        (+Sym)               (+Sym)

                                             K        Þ          K4

                                                     (+Trans)

Try                  []p É <>p in K.

                         We find that it doesn't close in any K* (non-reflexive normal) system.

The reason that tree doesn't close is because the initial world doesn't have access to any other world. 

This can be fixed following the Hintikka strategy by adding the following rule to K

[]D                      []X                w      

                           ...                              

                           <>X               w

Or, following the Orthodox strategy, by making the accessibility relation a serial one

Serial                []X                 w      

                           ...                              

                           wAv   

                         where v is new to this path

In either case we get the D logics: 

                                          DBr         Þ         D5

                                                    (+Trans)

                                             Ý                         Ý

                                        (+Sym)               (+Sym)

                     K        Þ         D         Þ          D4

                          (+Serial)         (+Trans)

You'll notice that K is a subsystem of all of these normal systems, so anything that is valid in K will be valid in any of the others.

Similarly, all of these systems are subsystems of S5, so anything that is invalid in S5 will be invalid in any of the others.